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Register a new userSparse dominations and weighted variation inequalities for singular integrals and commutators
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This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-H{o}rmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such variation operators.
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This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.
We study the commutators $[b,T]$ of pointwise multiplications and bi-parameter Calderon-Zygmund operators and characterize their off-diagonal $L^{p_1}L^{p_2} to L^{q_1}L^{q_2}$ boundedness in the range $(1,infty)$ for several of the mixed norm integrability exponents.
In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new $CMO(mathbb{R}^n)$ characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.
In this paper, the authors prove the boundedness of commutators generated by the weighted Hardy operator on weighted $lambda$-central Morrey space with the weight $omega$ satisfying the doubling condition. Moreover, the authors give the characterization for the weighted $lambda$-central Campanato space by introducing a new kind of operator which is related to the commutator of weighted Hardy operator.
We study the two-weighted estimate [ bigg|sum_{k=0}^na_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg|leq c|f|L_{p,u}(0,infty)|,tag{$*$} ] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<pleq qleqinfty$, provided that the weight $u$ satisfies the certain conditions, the estimate $(*)$ holds if and only if the estimate [ sum_{k=0}^nbigg|a_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg| leq c|f|L_{p,u}(0,infty)|.tag{$**$} ] is fulfilled. The necessary and sufficient conditions for $(**)$ to be valid are well-known. The obtained result can be applied to the estimates of differential operators with variable coefficients in some weighted Sobolev spaces.
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